Directivity of a distribution of baffled pistons with arbitrary phases

Using the first product theorem for arrays1 or the bridge theorem2, the directivity pattern of an arrangement of sources can be constructed by multiplying the directivity of one single source by a point source in the Fourier space.

Single baffled piston

A piston with radius $a$ is placed on a baffled plane

The directivity of the baffled piston is \begin{equation}\large D_p(\theta) = \frac{2J_1(ka\sin\theta)}{ka\sin\theta} \end{equation} and for different values of $ka$

Two pistons vibrating in phase and in anti-phase

The directivity for two pistons with relatively phase $\Phi_{12}$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 + e^{i(kd\sin\theta\cos\phi+\Phi_{12})}\right] \end{equation}

In phase

The directivity for when $\Phi_{12} =0$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 + e^{ikd\sin\theta\cos\phi}\right] \end{equation} and for different values of $ka$

In anti-phase

The directivity for when $\Phi_{12} =\pi$ is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\sin\theta\cos\phi}\right] \end{equation} and for different values of $ka$

Three Pistons on the plane in line

In this case, 2 pistons are vibrating with the same phase and the third in phase oppossition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{i2kd\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$

Three Pistons on the plane in "L"

In this case, 2 pistons are vibrating with the same phase and the third in phase-opposition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\alpha} - e^{ikd\beta}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$

Four pistons

Here, 2 pistons are vibrating with the same phase and the other two in phase opposition.

The directivity is \begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[-1 + e^{ikd\alpha} - e^{ik2d\alpha} + e^{ik3d\alpha}\right] \end{equation} where $\alpha = \sin\theta\cos\phi$. For different values of $ka$

Four pistons forming a squared

Again, two pistons are vibrating with the same phase and the other two in phase opposition.

\begin{equation}\large D(\theta,\phi) = D_p(\theta)\left[1 - e^{ikd\alpha} - e^{ikd\beta} + e^{i(kd\alpha + kd\beta)} \right] \end{equation}

where $\alpha = \sin\theta\cos\phi$ and $\beta = \sin\theta\sin\phi$. For different values of $ka$

References

[1] E. G. Williams, FOURIER ACOUSTICS Sound Radiation and Nearfield Acoustical Holography. 1999.

[2]L. L. Beranek and T. J. Mellow, Acoustics Sound Fields, Transducers and Vibration, Second edition. Academic Press, 2019. doi: 10.1016/B978-0-12-391421-7.00013-0.

Please refer to the paper : García A et al,Radiation efficiency of a distribution of baffled pistons with arbitrary phases, JASA, 2022 for more information.